For a prime number , the Prüfer -group is defined uniquely up to isomorphism as the group where every element has exactly roots. It is a divisible abelian group which can be described in several ways, for example:
It is , the colimit of the sequence of inclusions
As such, it is the initial algebra of the functor that pushes out along multiplication by .
The Prüfer -groups are the only infinite groups whose subgroups are totally ordered by inclusion. They are often useful as counterexamples in algebra; for example, a Prüfer group is an Artinian but not a Noetherian -module.